to replace \(\mathfrak {A}_{-}\)  in Eq.(REF ), it could be continuous across the bounce by properly choosing
\(\Theta (b)\) . In particular, the variable \(b(t)\)  satisfies Eq.(REF ) [1]}, [2]}, [3]} It is interesting to note that Eq.(REF ) holds not only in
mLQC-I, but also in LQC and mLQC-II.,
from which we can see that \(b(t)\)  is always a monotonically decreasing function for any matter that satisfies the weak energy condition [4]}.
Moreover, one can construct a step-like function of \(b\)  with the bounce as its symmetry axis [1]}, [2]}, [3]}. Therefore,
if we define \( \Theta (b)\)  as
\(\Theta (b) = 1-2\left(1+ \gamma ^2\right)\sin ^2\left(\lambda b\right),\) 
